The management of large complex systems is complicated by
self-regulation—the phenomenon whereby changes in one component lead to
compensating changes elsewhere. One characteristic of such systems (though by
no means inevitable) is the potential for chaos—an unstable dynamic that
makes predictions impossible. Papadopoulos et
al.1, in the
JRSM, and Smethurst and
Williams2, in
Nature, have argued that National Health Service (NHS) waiting lists
show self-regulatory behaviour and may be capable of chaotic behaviour. These
workers plotted the frequency distribution of the relative changes in queue
size on a double logarithmic scale. When the double logarithmic plots appeared
linear, the so-called power law was taken as evidence for self-regulation. We
believe this technique to be fundamentally flawed. Unregulated time series can
show power laws and we have suggested alternative
approaches3.

Here we apply appropriate tests to data on the size of waiting lists in 10
randomly chosen hospitals over the period 1998-2001 (data taken from
[www.doh.gov.ac.uk
]). To each dataset we applied three tests. First, to check for long-term
trends (e.g. increases or decreases in waiting lists), we performed linear
regressions of queue size on time. Second, we estimated slope of the relation
between change in log queue size and log queue size to see if this was
negative. This test looks for evidence that large queues tend to get smaller
whilst small queues get larger, as in self-regulated systems. Statistical
significance tests cannot be performed on the slopes of such regressions,
since for random time series the slope of this relation is biased and is
expected to be negative. Instead we were interested in whether slopes were
steeper than - 1, the implication of this value being that chaos is only
possible if the slope is steeper than - 1. Third, we tested for
self-regulation using a test (the Pollard test) explicitly designed to detect
self-regulation in time
series4.

The results are shown in Table
1. 6 out of 10 time series showed evidence for long-term trends,
all negative, indicating significant declines in queue length. The slopes
estimated from the regressions of change in log queue size on queue size were
steeper than - 1 only in one case, and this was only slightly steeper (-
1.11). As noted, statistical tests on these slopes are biased. The Pollard
test, which corrects for this bias, indicated that only one time series
exhibited statistically significant evidence for self-regulation; moreover, in
view of the number of tests (10), the marginal significance of this relation
(P=0.03) should be viewed with caution.

Analysis of data on waiting lists from 10 randomly chosen hospitals for
evidence of changes consistent with chaos or density dependence

We do not regard the analysis we have presented as definitive; for
instance, longer time series may show different behaviour. However, claims
that the NHS is ‘at the edge of chaos’ are not supported by our
analyses. We have shown that, for the data in question, there is no evidence
of self-regulatory behaviour. Moreover, waiting lists do show evidence of
clear declines, probably resulting from management strategies to reduce queue
sizes.